Precise coordinated planning over a forward time window enables safe and highly efficient motion when many robots must work together in tight spaces, but this would normally require centralised control of all devices which is difficult to scale. We demonstrate GBP Planning, a new purely distributed technique based on Gaussian Belief Propagation for multi-robot planning problems, formulated by a generic factor graph defining dynamics and collision constraints over a forward time window. In simulations, we show that our method allows high performance collaborative planning where robots are able to cross each other in busy, intricate scenarios. They maintain shorter, quicker and smoother trajectories than alternative distributed planning techniques even in cases of communication failure. We encourage the reader to view the accompanying video demonstration at https://youtu.be/8VSrEUjH610.